Notation scientific

Scientific notation uses exponents (powers of 10) for handling very large or very small numbers. A number in scientific notation consists of a number multiplied by a power of 10. The number is called the mantissa. In scientific notation, only one digit in the mantissa is to the left of the decimal place. The order of magnitude is expressed as a power of 10, and indicates how many places you had to move the decimal point so that only one digit remains to the left of the decimal point. 

Numbers in proper scientific notation have only one digit in front of the decimal place. Can you convert 12.5 x 10 3 into proper form You have to move the decimal point one place to the left, but does the exponent increase or decrease Is the correct answer 1.25 x 10-4 or 1.25 x 10-2 Rather than memorize a rule, think of it this way. You want your answer to be equal to the initial number. So, if you DECREASE 12.5 to 1.25, wouldn t you need to INCREASE the exponent to cancel out this change So, the correct answer is 1.25 x 10-2. 

You will probably use your calculator for most calculations. It is critical that you learn to use the scientific notation feature of your calculator properly. Calculators vary widely, but virtually all scientific calculators have either an EE key or an EXP key that is used for scientific notation. Unfortunately, these calculators also have a 10 key, which is the antilog key and has nothing to do with scientific notation, so don t use it when entering numbers in scientific 

Remember to follow the rules for symbols of enclosure. The division bar is a symbol of enclosure, so do the operations in the numerator and in the denominator before performing the division. By the way, did you notice you can easily solve this problem without a calculator Try regrouping the mantissas separately from the powers of 10. 

Scientific notation is frequently used in chemistry where we encounter both very small and very large quantities. Examples include a wavelength of 1.54 x 10 ° m and the value of Planck s constant 6.63 X10 J s. The common feature of these values is that they consist of a number between 1 and 10 multiplied by a power of 10. This is true of any number expressed in scientific notation. 

The value of n may also be negative. For example, 10 = 1/10 , as shown in Chapter 1. In this case we have 

It follows from the above that the number written as 

Suppose we have the number 8352. First we need to write the appropriate number between 1 and 10, which is 8.352. This number needs to be multiplied by 1000 or 10 to give the original value. So in scientific notation we write this as 8.352 x 10.  

Now consider 0.000 004 39. The appropriate number between 1 and 10 is 4.39. To give the original answer we need to multiply by 0.000001 or 10 . So in scientific notation this is given as 4.39 x 10- . 

Scientific notation, also known as exponential notation, is a way of representing large and small numbers as the product of two terms. The first term, the coefficient, is a number between 1 and 10. The second term, the exponential term, is 10 raised to a power—the exponent. For example. 

The exponent indicates the number of lO s hy which the coefficient is multiplied to give the number represented in scientific notation 

There are two great advantages to scientific notation. The first, as illustrated in Table B.l, is that the very large and very small numbers often dealt with in the sciences are much less cumbersome in scientific notation. The second is that it removes the ambiguity in the number of significant figures in a number ending with zeroes. We can, for example, write 

For numbers larger than 1, the exponent in scientific notation is a positive whole number, as illustrated above. For numbers less than 1, the exponent is a negative whole number that indicates the number of times the coefficient must be divided by 10 (or multiplied by 0.1) to give the number represented in scientific notation 

Do not use X 10 when entering exponential numbers on a calculator this multiplies your answer by 10. 

Every academic discipline has its own jargon. Chemistry is no exception. The purpose of this chapter is to introduce you to some of the terminology and mathematical notation used in chemistry. 

Science has its own language that scientists use to communicate with each other. Because scientists may be called upon to measure extremely large numbers (like the distances to far-flung galaxies) and extremely small numbers (hke the diameter of a proton), scientific notation is used. Scientists have also agreed to use units of measurement that make communication clearer. 

Generally speaking in science, the metric system is preferred, although nonmetric units may persist in areas where they are familiar to scientists working in that area. Most units have some shorthand form of abbreviation that scientists have been using for centuries. 

Consider the distance hght travels in a vacuum in one year, the distance astronomers call a light year. TVaveling at a constant speed of 186,000 miles/second for 31,558,000 seconds (one year), a Hght year is a distance of 5,869,713,600,000 miles. (Note A Ughtyear is close to 

Scientific notation refers to writing a number like 3,516 in the form 3-516 x 10, or 0.00052 as 5.2 X 10 . in scientific notation, a number is written as a single digit, a decimal point, and any 

Scientists often use numbers that are very large or very small in measurements. For example, the Earth s age is estimated to be about 4,500,000 (4.5 billion) years. Numbers like these are bulky to write, so to make them more compact scientists use powers of 10. Writing a number as the product of a number between 1 and 10 multiplied by 10 raised to some power is called scientific notation. 

To learn how to write a number in scientific notation, let s consider the number 2468. To write this number in scientific notation  

Move the decimal point in the original number so that it is located after the first nonzero digit. 

Multiply this new number by 10 raised to the proper exponent (power). The proper exponent is equal to the number of places that the decimal point was moved. 

The sign on the exponent indicates the direction the decimal was moved. 

As we will discuss in this chapter, a measurement always consists of two parts a number and a unit. Both parts are necessary to make the measurement meaningful. For example, suppose a friend tells you that she saw a bug 5 long. This statement is meaningless as it stands. Five what If it s 5 millimeters, the bug is quite small. If it s 5 centimeters, the bug is quite large. If it s 5 meters, run for cover  

The point is that for a measurement to be meaningful, it must consist of both a number and a unit that tells us the scale being used. 

In this chapter we will consider the characteristics of measurements and the calculations that involve measurements. 

A measurement must always consist of a number and a unit. 

AIM To show how very large or very small numbers can be expressed as the product of a number between 1 and 10 and a power of 10. 

English system Metric system International System (SI) 

When writing a number in scientific notation, keep one digit to the left of the decimal point. 

What is the difference between a qualitative observation and a quantitative observation  

In science, we often encounter extremely large numbers and extremely small numbers. For example, in Chapter 7 we will encounter a number that has 20 zeros after the last nonzero digit before the decimal point is found. The number is 

Moving the decimal just to the right of the first digit, the correct way to express this number in scientific notation then is as follows  

The exponent of the 10 in this notation is the number of places we moved the decimal point. Expressing Avogadro s number in scientific notation, we would have the following number  

Engineers must frequently perform calculations using extremely large or very small numbers. In such instances, it may become cumbersome to write such numbers in decimal form. For example, consider the following multiplication operation  

Notice that multiplication of numbers with powers of 10 is performed by adding the exponents algebraically. Furthermore, the product is recorded to two significant digits, the fewest significant digits of numbers used in the multiphcation. 

With scientific notation, the exponents of 10 are used to indicate the decimal place. Thus, 

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Chemists frequently work with measurements that are very large or very small. A mole, for example, contains 602,213,670,000,000,000,000,000 particles, and some analytical techniques can detect as little as 0.000000000000001 g of a compound. For simplicity, we express these measurements using scientific notation thus, a mole contains 6.0221367 X 10 particles, and the stated mass is 1 X 10 g. Sometimes it is preferable to express measurements without the exponential term, replacing it with a prefix. A mass of 1 X 10 g is the same as 1 femtogram. Table 2.3 lists other common prefixes. 

Real—Predefined scalar type, decimal, or scientific notation may be used. Boolean—Predefined logical type, ordered so that false < true. 

In general, any ambiguity concerning the number of significant figures in a measurement can be resolved by using exponential notation (often referred to as scientific notation ), discussed in Appendix 3. 

Numbers such as these are very awkward to work with. For example, neither of the numbers just written could be entered directly on a calculator. Operations involving very large or very small numbers can be simplified by using exponential (scientific) notation. To express a number in exponential notation, write it in the form... 

Schrodinger equation, 140 Schrodinger, Erwin, 139 Scientific notation, 10 Seaborg, Glenn, 147,174,515... 

In scientific notation, a number is written as A x 10 7. Here A is a decimal number with one nonzero digit in front of the decimal point and a is a whole number. For example, 333 is written 3.33 X 102 in scientific notation, because 102= 10x 10 = 100 ... 

To multiply numbers in scientific notation, multiply the decimal parts of the numbers and add the powers of 10 ... 

When raising a number in scientific notation to a particular power raise the decimal part of the number to the power and multiply the power of 10 by the power ... 

Schilling test, 727 Schrodinger, E., 16 Schrodinger equation, 17 scientific method, F2 scientific notation, AS scintillation counter, 711 sea of instability, 705 second, F6, A3 second derivative, A9 second ionization energy, 43 second law of... 

As usual, Feynman was right. His little particles captures an essential fact about atoms. They are tiny—so tiny that a teaspoon of water contains about 500,000,000,000,000,000,000,000 of them. Handling numbers this big is awkward. Try dividing it by 63, for example. To accommodate the very large numbers encountered in counting atoms and the very small ones needed to measure them, chemists use the scientific notation system. 

Scientific notation uses exponents to express numbers. The number 1,000, for instance, is equal to 10 x 10 x 10, or 10. The number of zeros following the 1 in 1,000 is 3, the same as the exponent in scientific notation. Similarly, 10,000, with 4 zeros, would be 10 , and so on. The same rules apply to numbers that are not even multiples of 10. For example, the number 1,360 is 1.36 x 10. And the number of atoms in a spoonful of water becomes an easy-to-write 5 X 10. ... 

Scientific notation is also useful for representing very small numbers. The number 0.1 would be 1/10 or 10". The radius of an atom of aluminum is 0.000000000143 meters. Using scientific notation, we could write this distance more compactly as 1.43 x 10 °. Atoms are, indeed, very little particles. ... 

The abbreviation log stands for logarithm. In mathematics, a logarithm is the power (also called an exponent) to which a number (called the base) has to be raised to get a particular number. In other words, it is the number of times the base (this is the mathematical base, not a chemical base) must be multiplied times itself to get a particular number. For example, if the base number is 10 and 1,000 is the number trying to be reached, the logarithm is 3 because 10 x 10 x 10 equals 1,000. Another way to look at this is to put the number 1,000 into scientific notation ... 

When scientists write numbers in exponential form, they prefer to write them so that the coefficient has one and only one digit to the left of the decimal point. That notation is called standard exponential form or scientific notation. 

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